On the determination of the stochasticity threshold of invariant curves

We consider the problem of determining the stochasticity transition value in nearly-integrable mappings. We perform explicitly a canonical transformation, which conjugates the original mapping to an integrable one, up to a given order in the perturbing parameter. Then we derive a numerical evidence of the existence of an invariant curve associated with the transformed system and, correspondingly, to the original one. In the second part of the paper we implement a numerical method due to M. Hénon [Hénon] for the computation of the rotation number corresponding to a given initial condition. Following an idea of Laskar et al. [1992] and Laskar [1993], we determine with high accuracy the critical breakdown threshold of invariant curves for standard-mapping like systems which allows not only to test Hénon's method but also to compare our analytical results with an accurate numerical one. An application is also made about the accuracy of the leap frog method.